Abstract
In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.
Highlights
Piecewise-smooth (PWS) dynamical models have become in valuable tools to analyze many physical systems [1]
We propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations in n-dimensional PWS dynamical systems, Continuous PWS and Discontinuous PWS [9]
These bifurcations were not characterized completely due to the absence of a framework that allows to finding differences between nonsmooth cycles caused by multiple discontinuity boundaries (DBs) or corner manifolds (CM)
Summary
Piecewise-smooth (PWS) dynamical models have become in valuable tools to analyze many physical systems [1]. The model of a non-smooth rotational oscillator in contact with one or two different rough discs rotating with constant driving velocities is considered [15] and the model of double-belt friction oscillator (DBO) is proposed [16] Both works describe non-standard bifurcations that occur by influence of several DBs. Both works describe non-standard bifurcations that occur by influence of several DBs These bifurcations were not characterized completely due to the absence of a framework that allows to finding differences between nonsmooth cycles caused by multiple DBs or CMs. For example, the right side of Figure 2 presents a nonsmooth limit cycle of a mechanical oscillator with double cam [14]. Discontinuity-Induced Bifurcations (DIBs) of nonsmooth limit cycles in PWS dynamical systems.
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